SSAT Middle Level Quantitative - How to find the area of a triangle

What is the area of the triangle?

Question_11

$\small 35$

$\small 70$

$\small 84$

$\small 42$

Explanation

Area of a triangle can be determined using the equation:

$\small A=\frac{1}{2}bh$

$\small A=\frac{1}{2}(14)(5)=35$

Rectangles

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the ratio of the area of the green region to that of the white region.

The correct answer is not given among the other choices.

Explanation

The area of the entire rectangle is the product of its length and width, or

$120 \times 50 = 6,000$ .

The area of the right triangle is half the product of its legs, or

$\frac{1}{2} \times 40 \times 80 = 1,600$

The area of the green region is therefore the difference of the two, or

The ratio of the area of the green region to that of the white region is

$\frac{4,400}{1,600} = \frac{4,400 \div 400}{1,600\div 400} = \frac{11}{4}$

That is, 11 to 4.

Rectangles

Note: Figure NOT drawn to scale.

What percent of the above figure is green?

$73 \frac{1}{3} %$

$62\frac{1}{2} %$

$78 \frac{3}{4} %$

$68 \frac{8}{9} %$

The correct answer is not given among the other choices.

Explanation

The area of the entire rectangle is the product of its length and width, or

$120 \times 50 = 6,000$ .

The area of the right triangle is half the product of its legs, or

$\frac{1}{2} \times 40 \times 80 = 1,600$

The area of the green region is therefore the difference of the two, or

The green region is therefore

$\frac{4,400}{6,000} \times 100 = 73 \frac{1}{3} %$

of the rectangle.

Rectangles 3

The above diagram shows Rectangle , with midpoint of $\overline{CT}$ .

The area of $\bigtriangleup ECM$ is 225. Evaluate

Explanation

is the midpoint of $\overline{CT}$ , so $\bigtriangleup ECM$ has as its base $CM= \frac{1}{2} \cdot CT = \frac{1}{2} \cdot 2x =x$ ; its height is $EC = \frac{1}{2} x$ .

Its area is half their product, or

$\frac{1}{2} \cdot x \cdot \frac{1}{2} x = \frac{1}{4} x^{2}$

Set this equal to 225:

$\frac{1}{4} x^{2} = 225$

$4 \cdot \frac{1}{4} x^{2} = 4 \cdot 225$

$x^{2} = 900$

$x = \sqrt{900}= 30$ .

Right triangle 2

Give the perimeter of the above triangle in feet.

$2 \frac{1}{2}\textrm{ ft }$

$2 \frac{2}{3}\textrm{ ft }$

$3 \frac{1}{2}\textrm{ ft }$

$3 \frac{1}{3}\textrm{ ft }$

Explanation

The perimeter of the triangle - the sum of the lengths of its sides - is

inches.

Divide by 12 to convert to feet:

$30 \div 12 = 2 \textrm{ R }6$

As a fraction, this is $2 \frac{6}{12}$ or $2 \frac{1}{2}$ feet,

A triangle has a height of 9 inches and a base that is one third as long as the height. What is the area of the triangle, in square inches?

$\frac{27}{2}$

None of these

Explanation

The area of a triangle is found by multiplying the base times the height, divided by 2.

$\text{Area} =\frac{\text{base}\times \text{height}}{2}$

Given that the height is 9 inches, and the base is one third of the height, the base will be 3 inches.

$b=h\div3=9\div3=3$

We now have both the base (3) and height (9) of the triangle. We can use the equation to solve for the area.

$\text{Area} =\frac{9\times 3}{2}$

$\text{Area} =\frac{27}{2}$

The fraction cannot be simplified.

Square

The quadrilateral in the above diagram is a square. What percent of it is white?

$14 \frac{1}{16} %$

$16 \frac{1}{4} %$

$28 \frac{1}{8} %$

$17 \frac{1}{2} %$

Explanation

The area of the entire square is the square of the length of a side, or

$80 \times 80 = 6,400$ .

The area of the white right triangle is half the product of its legs, or

$\frac{1}{2} \times 30 \times 60 = 900$ .

Therefore, the area of that triangle is

$\frac{900 }{6,400} \times 100 = 14 \frac{1}{16} %$

of that of the entire square.

The hypotenuse of a right triangle is feet; it has one leg feet long. Give its area in square inches.

$38,880 \textrm{ in}^{2}$

$77,760 \textrm{ in}^{2}$

$101,088 \textrm{ in}^{2}$

$42,120 \textrm{ in}^{2}$

$43,120 \textrm{ in}^{2}$

Explanation

The area of a right triangle is half the product of the lengths of its legs, so we need to use the Pythagorean Theorem to find the length of the other leg. Set :

$a^{2} = c^{2} - b^{2}$

$a^{2} = 39 ^{2} - 36^{2}$

$a^{2} = 1,521 - 1,296$

$a^{2} = 225$

$a = \sqrt{225} = 15$

The legs have length and feet; multiply both dimensions by to convert to inches:

$15 \times 12 = 180$ inches

$36 \times 12 = 432$ inches.

Now find half the product:

$A = \frac{1}{2} \cdot 180 \cdot 432 = 38,880\ in^{2}$

Yard_2

Mr. Jones owns the isosceles-triangle-shaped parcel of land seen in the above diagram. He sells the parcel represented in red to his brother. What is the area of the land he retains?

$7,616\textrm{ m}^{2}$